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On groupoids with involutions and their cohomology
Elkaioum Moutuou
To cite this version:
Elkaioum Moutuou. On groupoids with involutions and their cohomology. New York Journal of Mathematics, Electronic Journals Project, 2013. �hal-01289008�
New York Journal of Mathematics
New York J. Math. 19 (2013) 729–792.
On groupoids with involutions and their
cohomology
El-ka¨ıoum M. Moutuou
Abstract. We extend the definitions and main properties of graded ex-tensions to the category of locally compact groupoids endowed with in-volutions. We introduce Real ˇCech cohomology, which is an equivariant-like cohomology theory suitable for the context of groupoids with invo-lutions. The Picard group of such a groupoid is discussed and is given a cohomological picture. Eventually, we generalize Crainic’s result, about the differential cohomology of a proper Lie groupoid with coefficients in a given representation, to the topological case.
Contents
0. Introduction 730
1. Real groupoids and Real graded extensions 731
1.1. Real groupoids 731
1.2. RealG-bundles 734
1.3. Generalized morphisms of Real groupoids 735
1.4. Morita equivalence 737
1.5. Real graded twists 743
1.6. Real graded central extensions 747
1.7. Functoriality of dExtR(·, S) 749
2. Real ˇCech cohomology 751
2.1. Real simplicial spaces 751
2.2. Real sheaves on Real simplicial spaces 754 2.3. Real G-sheaves and reduced Real sheaves 758
2.4. Real G-modules 761
2.5. Pre-simplicial Real covers 762
2.6. “Real” ˇCech cohomology 763
2.7. Comparison with usual groupoid cohomologies 770
2.8. The group ˇHR0 771
Received April 29, 2012; revised October 23, 2013.
2010 Mathematics Subject Classification. 22A22; 55N32; 53C08.
Key words and phrases. Real groupoids, groupoid cohomology, graded extensions. Partially suported by the German Research Foundation (DFG) via the IRTG 1133.
ISSN 1076-9803/2013
2.9. HRˇ 1 and the Real Picard group 772
2.10. HRˇ 2 and ungraded Real extensions 776
2.11. The cup-product ˇHR1(·, Z2) × ˇHR1(·, Z2) → ˇHR2(·, S1) 781
2.12. Cohomological picture of the group dExtR(G, S1) 782
2.13. The proper case 783
Acknowledgements 790
References 790
0. Introduction
A Real1 object in a category C is a pair (A, f) consisting of an object A ∈ Ob(C) together with an element f ∈ IsomC(A, A), called the Real struc-ture, such that f2 = 1A. For instance, an Atiyah Real space (X, τ ) [2] is
nothing but a Real object in the category of locally compact spaces. We are particularly interested in the category Gs [25] of locally compact Hausdorff
groupoids with strict homomorphisms [15, 16] as morphisms; we shall refer to Real objects in Gs as Real groupoids. For example, let WPn(a1,...,an) be
the weighted projective orbifold [1] associated to the pairwise coprime inte-gers a1, . . . , an; then together with the coordinate-wise complex conjugation,
WPn(a1,...,an) is a Real groupoid.
A morphism of Real groupoids is a morphism in Gs intertwining the
Real structures. We may also speak of a Real strict homomorphism. Real groupoids form a category RGs in which morphisms are Real strict
ho-momorphisms. Moreover, they are the objects of a 2-category RG(2) de-fined as follows. Let (G, ρ), (Γ, %) ∈ Ob(RGs). A generalized
homomor-phism [7, 9, 16, 25] Γ−→Z G is said to be Real if Z is given a Real structure τ such that the moment maps and the groupoid actions respect some co-herent compatibility conditions with respect to the Real structures. A mor-phism of Real generalized homomormor-phisms (Z, τ ) −→ (Z0, τ0) is a morphism of generalized homomorphisms Z −→ Z0 intertwining the Real structures. Henceforth, 1-morphisms in RG(2) are Real generalized homomorphisms and 2-morphisms are morphisms of Real generalized homomorphisms. All functorial properties we deal with in this paper are however discussed in the category RG defined as RG(2) “up to 2-isomorphisms”.
In [21], a ˇCech cohomology theory for topological groupoids is defined as the ˇCech cohomology of simplicial topological spaces, and it is shown that the well-known isomorphism between S1-central extensions of a discrete groupoid G and the second cohomology group [19, 11] of G with coefficients in the sheaf of germs of S1-valued functions also holds in the general case; i.e., Ext(G, S1) ∼= ˇH2(G
•, S1). We define here an analogous theory ˇHR∗ 1
Note the capitalization, used to avoid confusion with a module over R or a real manifold.
that fits well the context of Real groupoids. This theory was motivated by the classification of groupoid C∗-dynamical systems endowed with involu-tions [17]. These can be thought of as a generalization of continuous-trace C∗-algebras with involutions. Specifically, it is known [20] that given such a C∗-algebra A, its spectrum X admits a Real structure τ , and its Dixmier– Douady invariant δ(A) ∈ ˇH2(X, S1) is such that δ(A) = τ∗δ(A), where the “bar” is the complex conjugation in S1. In fact, thinking of X as a Real groupoid, we will see that all 2-cocycles satisfying the latter relation are classified by ˇHR2(X, S1), where S1 is endowed with the complex conjuga-tion. ˇHR∗ appears then to provide the right cohomological interpretation of C∗-dynamical systems with involutions.
We try, to the extent possible, to make the present paper self-contained. We start by collecting, in Section 1, a number of notions and results about Real groupoids most of which are adapted from many sources in the litera-ture [15, 19, 25]; specifically, we define the group [ExtR(G, S) of (equivalence classes of) Real graded S-central extensions over a Real groupoid G, by a given Real abelian group S. In Section 2, we introduce Real ˇCech cohomol-ogy, following closely [21]. While ˇHR∗ behaves almost like a Z2-equivariant
cohomology theory, we will see that it is actually not. Geometric interpre-tations of the cohomology groups ˇHR1(G•, S) and ˇHR2(G•, S), for a Real
Abelian group S, are given. Finally, we generalize a result by Crainic [4] (on the differential cohomology groups of a proper Lie groupoid) to topological proper (Real) groupoid.
1. Real groupoids and Real graded extensions
Recall [19, 16, 25] that a strict homomorphism between two groupoids G ////X and Γ ////Y is a functor ϕ : Γ −→ G given by a map Y −→ X on objects and a map Γ(1) −→ G(1) on arrows, both denoted
again by ϕ, which preserve the groupoid structure maps, i.e., ϕ(s(γ)) = s(ϕ(γ)), ϕ(r(γ)) = r(ϕ(γ)), ϕ(1y) = 1ϕ(y) and ϕ(γ1γ2) = ϕ(γ1)ϕ(γ2)
(hence ϕ(γ−1) = ϕ(γ)−1), for all (γ1, γ2) ∈ Γ(2)and y ∈ Y . Unless otherwise
specified, all our groupoids are topological groupoids which are supposed to be Hausdorff and locally compact.
1.1. Real groupoids.
Definition 1.1. A Real groupoid is a groupoid G ////X together with a strict 2-periodic homeomorphism ρ : G −→ G. The homeomorphism ρ is called a Real structure on G. Such a groupoid will be denoted by a pair (G, ρ).
Example 1.2. Any topological Real space (X, ρ) in the sense of Atiyah [2] can be viwed as a Real groupoid whose the unit space and the space of morphisms are identified with X; i.e., the operations in this Real groupoid is defined by s(x) = r(x) = x, x · x = x, x−1 = x.
Example 1.3. Any group with involution can be viewed as a Real groupoid with unit space identified with the unit element. Such a group will be called Real.
Lemma 1.4. Let G be an abelian group equipped with an involution τ : G −→ G ( i.e., a Real structure). Set
<(τ ) := {g ∈ G | τ (g) = g} =RG, =(τ ) := {g ∈ G | τ (g) = −g}. Then, (1.1) G ⊗ Z 1 2 ∼ = (<(τ ) ⊕ =(τ )) ⊗ Z 1 2 .
If τ is understood, we will write IG for =(τ ). We call <(τ ) and =(τ ) the Real part and the imaginary part of G, respectively.
Proof. For all g ∈ G, one has g + τ (g) ∈RG, and g − τ (g) ∈IG. Therefore,
after tensoring G with Z[1/2], every g ∈ G admits a unique decomposition g = g + τ (g) 2 + g − τ (g) 2 ∈ Z[1/2] ⊗ RG ⊕IG.
Example 1.5. Let n ∈ N∗. Suppose ρ is a Real structure on the additive group Rn. Then there exists a unique decomposition Rn = Rp⊕ Rq such
that ρ is determined by the formula
ρ(x, y) = (1p⊕ (−1q))(x, y) := (x, −y),
for all (x, y) = (x1, · · · , xp, y1, · · · , yq) ∈ Rp⊕ Rq.
For each pair (p, q) ∈ N, we will write Rp,q for the additive group Rp+q equipped with the Real structure (1p⊕ (−1q)).
Define the Real space Sp,q as the invariant subset of Rp,q consisting of elements u ∈ Rp+q of norm 1. For q = p, Sp,p is clearly identified with the
Real space Sp whose Real structure is given by the coordinate-wise complex conjugation. Notice that rSp,q= Sp,0.
Example 1.6. Let (X, ρ) be a topological Real space. Consider the fun-damental groupoid π1(X) over X whose arrows from x ∈ X to y ∈ X are
homotopy classes of paths (relative to end-points) from x to y and the par-tial multiplication given by the concatenation of paths. The involution ρ induces a Real structure on the groupoid as follows: if [γ] ∈ π1(X), we set
ρ([γ]) the homotopy classes of the path ρ(γ) defined by ρ(γ)(t) := ρ(γ(t)) for t ∈ [0, 1].
Two Real structures ρ and ρ0 on G are said to be conjugate if there exists a strict homeomorphism φ :G −→ G such that ρ0= φ ◦ ρ ◦ φ−1. In this case we say that the Real groupoids (G, ρ) and (G, ρ0) are equivalent.
Definition 1.7. We write rG ////rX (orρG when there is a risk of
Lemma 1.8. Let G and Γ be Real groupoids, and let φ : Γ −→ G be a Real groupoid homomorphism, then φ(rΓ) is a full subgroupoid of rG ////rX .
If in addition φ is an isomorphism, then rΓ ∼= rG ////rX .
In particular, if ρ1 and ρ2 are two conjugate Real structures on G, then ρ1G ∼=ρ2G.
Proof. This is obvious since φ(¯γ) = φ(γ) for all γ ∈ Γ. Remark 1.9. Note that the converse of the second statement of the above lemma is false in general. For instance, consider the Real group S1 whose
Real structure is given by the complex conjugation, and the Real group Z2
(with the trivial Real structure). We have rS1= {±1} ∼= Z2=rZ2.
The following is an example of groupoids with equivalent Real structures. Example 1.10. Recall ([8, IV.3]) that a Riemannian manifold X is called globally symmetric if each point x ∈ X is an isolated fixed point of an involutory isometry sx : X −→ X; i.e., sx is a diffeomorphism verifying
s2
x = IdX and sx(x) = x. Moreover, for every two points x, y ∈ X, sx
and sy are related through the formula sx◦ sy ◦ sx = ssx(y). Given such
a space, each point x ∈ X defines a Real structure on X which leaves x fixed. However, let x and y be two different points in X and let z ∈ X be such that y = sz(x). Then, we get sz◦ sx ◦ sz = sy which means that
the diffeomorphism sz : X −→ X implements an equivalence sx ∼ sy. But
since x and y are arbitrary, it turns out that all of the Real structures sx
are equivalent. Thus, all of the Real spaces (X, sx) are equivalent to each
others.
Now, recall [8, IV. Theorem 3.3] that if G denotes the identity component of I(X), where the latter is the group of isometries on X, then the map σx0 : g 7−→ sx0gsx0 is an involutory automorphism in G, for any arbitrary
x0 ∈ X. It follows that all of the points of X give rise to equivalent Real
groups (G, σx).
From now on, by a Real structure on a groupoid, we will mean a represen-tative of a conjugation class of Real structures. Moreover, we will sometimes put ¯g := ρ(g), and writeG instead of (G, ρ) when ρ is understood.
Definition 1.11 (Real covers). Let (X, ρ) be a Real space. We say that an open cover U = {Ui}i∈I of X is Real if U is invariant with respect to
the Real structure ρ; i.e., ρ(Ui) ∈ U, ∀i ∈ I. Alternatively, U is Real if I is
equipped with an involution i 7−→ ¯i such that U¯i= ρ(Ui) for all i ∈ I.
Remark 1.12. Observe that Real open covers always exist for all locally compact Real space X. Indeed, let V = {Vi0}i0∈I0 be an open cover of the
space X. Let I := I0 × {±1} be endowed with the involution (i0, ±1) 7−→
(i0, ∓1). Next, put U(i0,±1):= ρ(±1)(Vi0), where ρ(+1)(g) := g, and ρ(−1)(g) :=
Definition 1.13 (Real action). Let (Z, τ ) be a locally compact Hausdorff Real space. A (continuous) right Real action of (G, ρ) on (Z, τ) is given by a continuous open map s : Z −→ X (called the generalized source map) and a continuous map Z ×s,X,rG −→ Z, denoted by (z, g) 7−→ zg, such that:
(a) τ (zg) = τ (z)ρ(g) for all (z, g) ∈ Z ×s,X,r G.
(b) ρ(s(z)) = s(τ (z)) for all z ∈ Z. (c) s(zg) = s(g).
(d) z(gh) = (zg)h for (z, g) ∈ Z ×s,X,rG and (g, h) ∈ G(2).
(e) zs(z) = z for any z ∈ Z where we identify s(z) with its image in G by the inclusion X ,→G.
If such a Real action is given, we say that (Z, τ ) is a (right) Real G-space. Likewise a (continuous) left Real action of (G, ρ) on (Z, τ) is determined by a continuous Real open surjection r : Z −→ X (the generalized range map of the action) and a continuous Real map G ×s,X,rZ −→ Z satisfying
the appropriate analogues of conditions (a), (b), (c), (d) and (e) above. Given a right Real action of (G, ρ) on (Z, τ) with respect to s, let
Ψ : Z ×s,X,rG −→ Z × Z
be defined by the formula Ψ(z, g) = (z, zg). Then we say that the action is free if this map is one-to-one (or in other words if the equation zg = z implies g = s(z). The action is called proper if Ψ is proper.
Notations 1.14. If we are given such a right (resp. left) Real action of (G, ρ) on (Z, τ), and if there is no risk of confusion, we will write Z ∗ G (resp. G ∗ Z) for Z ×s,X,rG (resp. for G ×s,X,rZ).
1.2. Real G-bundles.
Definition 1.15. Let (G, ρ) be a Real groupoid. A Real (right) G-bundle over a Real space (Y, %) is a Real (right)G-space (Z, τ) with respect to a map s: Z −→ X, together with a Real map π : Z −→ Y satisfying the relation π(zg) = π(z) for any (z, g) ∈ Z ×s,X,r G, and such that for any y ∈ Y , the
induced map
τy : Zy −→ Z%(y)
on the fibres isG-antilinear in the sense that for (z, g) ∈ Zy×s,X,rG we have
τy(zg) = τy(z)ρ(g)
as an element in Z%(y).
Such a bundle (Z, τ ) is said to be principal if:
(i) π : Z −→ Y is locally split (i.e., it is surjective and admits local sections).
(ii) The map Z ×s,X,rG −→ Z ×Y Z, (z, g) 7−→ (z, zg) is a Real
Remarks 1.16.
(1) The unit bundle. Given a Real groupoid (G, ρ), its space of arrows G(1) is a G-principal Real bundle over X. Indeed, the projection is
the range map r :G(1)−→ X, the generalized source map is given by s and the action is just the partial multiplication onG. This bundle is denoted by U (G) and is called the unit bundle of G (see [16]). (2) Pull-back. Let Z s // π X Y
be aG-principal Real bundle and f : Y0 −→ Y be a Real continuous map. Then the pull-back f∗Z := Y0×Y Z equipped with the invo-lution (%0, τ ) has the structure of aG-principal Real bundle over Y0. Indeed, the right Real G-action is given by the G-action on Z and the generalized source map is s0(y0, z) := s(z).
(3) Trivial bundles. From the previous two remarks, we see that if (Z, τ ) is any Real space together with a Real map ϕ : Z −→ X, then we get aG-principal Real bundle ϕ∗U (G) over Z; its total space being the space Z ×ϕ,X,rG. A Bundle of this form is called trivial
while aG-principal Real bundle which is locally of this form is called locally trivial.
1.3. Generalized morphisms of Real groupoids.
Definition 1.17. A generalized morphism from a Real groupoid (Γ, %) to a Real groupoid (G, ρ) consists of a Real space (Z, τ), two maps
Y oo r Z s //X ,
a left (Real) action of Γ with respect to r, a right (Real) action of G with respect to s, such that:
(i) The actions commute, i.e., if (z, g) ∈ Z ×s,X,rG and (γ, z) ∈ Γ×s,Y,rZ
we must have s(γz) = s(z), r(zg) = r(z) so that γ(zg) = (γz)g. (ii) The maps s and r are Real in the sense that s(τ (z)) = ρ(s(z)) and
r(τ (z)) = %(r(z)) for any z ∈ Z.
(iii) r : Z −→ Y is a locally trivialG-principal Real bundle.
Example 1.18. Let f : Γ −→G be a Real strict morphism. Let us consider the fibre product Zf := Y ×f,X,rG and the maps r : Zf −→ Y, (y, g) 7−→ y
and s : Zf −→ X, (y, g) 7−→ s(g). For (γ, (y, g)) ∈ Γ ×s,Y,rZf), we set
γ.(y, g) := (r(γ), f (γ)g) and for ((y, g), g0) ∈ Zf ×s,X,r G we set (y, g).g0 :=
(y, gg0). Using the definition of a strict morphism, it is easy to check that these maps are well-defined and make Zf into a generalized morphism from
Γ to G. Furthermore, the map τ on Zf defined by τ (y, g) := (%(y), ρ(g)) is
Definition 1.19. A morphism between two such morphisms (Z, τ ) and (Z0, τ0) is a Γ-G-equivariant Real map ϕ : Z −→ Z0 such that s = s0◦ ϕ and r = r0◦ ϕ. We say that the Real generalized homomorphism (Z, τ ) and (Z0, τ0) are isomorphic if there exists such a ϕ which is at the same time a homeomorphism.
Compositions of Real generalized morphisms are defined by the following proposition.
Proposition 1.20. Let (Z0, τ0) and (Z00, τ00) be Real generalized homomor-phisms from (Γ, %) to (G0, ρ0) and from (G0, ρ0) to (G, ρ) respectively. Then
Z = Z0×G0 Z00:= (Z0×
s0,G0(0),r00Z00)/(z0,z00)∼(z0g0,g0−1z00)
with the obvious Real involution, defines a Real generalized morphism from Γ ////Y to G ////X .
Proof. Let us first describe the structure maps
Y oo r Z s //X and the actions.
For (z0, z00) ∈ Z we set r(z0, z00) := r0(z0) and s(z0, z00) := s00(z00). These are well-defined and since
s(z0g0, g0−1z00) = s00(g0−1z00) = s00(z00), r(z0g0, g0−1z00) = r0(z0g0) = s0(z0),
from Definition 1.17(i). The actions are defined by γ.(z0, z00) := (γz0, z00) and (z0, z00).g := (z0, z00g) for (γ, (z0, z00)) ∈ Γ ×s,Y,rZ and ((z0, z00), g) ∈ Z ×s,X,rG
while the Real involution is the obvious one: τ (z0, z00) := (τ0(z0), τ00(z00)).
Now to show the local triviality of Z, notice that from (3) of Remarks 1.16, Z0 and Z00are locally of the form U ×ϕ0,G0(0),r0G0and V ×ϕ00,X,rG respectively,
where ϕ0 : U −→G0(0) and ϕ00 : V −→ X are Real continuous maps, U and V subspaces of Y and G0(0) respectively. It turns out that by construction, Z is locally of the form W ×ϕ,G0(0),rG where W = U ×ϕ0,G0(0) V .
Definition 1.21. Given two Real generalized morphisms (Γ, %)(Z,τ )−→ (G0, ρ0) and (G0, ρ0)(Z
0,τ0)
−→ (G, ρ), we define their composition (Z0◦ Z, τ ) : (Γ, %) −→ (G, ρ) to be (Z ×G0Z0, τ × τ0).
Remark 1.22. It is easy to check that the composition of Real generalized homomorphisms is associative. For instance, if
Γ(Z1,ρ1)//G1 (Z2,ρ2)//
G2 (Z3,ρ3)//
are given Real generalized morphisms, we get two Real generalized mor-phisms Z = Z1 ×G1 (Z2 ×G2 Z3) and Z0 = (Z1 ×G1 Z2) ×G2 Z3 between
(Γ, %) and (G, ρ); notice that here Z and Z0 carry the obvious Real invo-lutions. Moreover, the map Z −→ Z0, (z1, (z2, z3)) 7−→ ((z1, z2), z3) is a
Γ-G-equivariant Real homeomorphism. Hence, there exists a category RG whose objects are Real locally compact groupoids and morphisms are iso-morphism classes of Real generalized homoiso-morphisms.
Lemma 1.23. Let f1, f2 : Γ →G be two Real strict homomorphisms. Then
f1 and f2 define isomorphic Real generalized homomorphisms if and only if
there exists a Real continuous map ϕ : Y −→G such that f2(γ) = ϕ(r(γ))f1(γ)ϕ(s(γ))−1.
Proof. Le Φ : Zf1 −→ Zf2 be a Real Γ-G-equivariant homeomorphism,
where Zfi = Y ×fi,X,rG. Then from the commutative diagrams
Y Zf1 pr1 oo s◦pr2 // Φ X Zf2 pr1 __ s◦pr2 >>
we have Φ(x, g) = (x, h) with s(g) = s(h); and then there exists a unique element ϕ(x) ∈G such that h = ϕ(x)g. To see that this defines a continuous map ϕ : Y −→G, notice that for any x ∈ Y , the pair (x, f1(x)) is an element
in Zf1, then ϕ(x) is the unique element in G such that
Φ(x, f1(x)) = (x, ϕ(x)f1(x)).
Furthermore, since Φ is Real,
Φ(%(x), ρ(f1(x))) = (%(x), ρ(ϕ(x))ρ(f1(x))),
which shows that ϕ(%(x)) = ρ(ϕ(x)) for any x ∈ Y ; i.e., ϕ is Real.
Now for γ ∈ Γ, take x = s(γ), then from the Γ-equivariance of Φ, we have Φ(γ · (s(γ), f1(s(γ)))) = Φ(r(γ), f1(γ)) = γ · Φ(s(γ), f1(s(γ)));
so that
(r(γ), ϕ(r(γ))f1(γ)) = (r(γ), f2(γ)ϕ(s(γ)))
and f2(γ) · r(ϕ(s(γ))) = ϕ(r(γ))f1(γ)ϕ(s(γ)); but r(ϕ(s(γ))) = s(f2(γ)) by
definition of ϕ and this gives the desired relation.
The converse is easy to check by working backwards. 1.4. Morita equivalence. Let (Γ, %) and (G, ρ) be two Real groupoids. Suppose that f : (Γ, %) −→ (G, ρ) is an isomorphism in the category RGs.
In this case, we say that (Γ, %) and (G, ρ) are strictly equivalent and we write (Γ, %) ∼strict (G, ρ). Now, consider the induced Real generalized morphisms
(Zf, τf) : (Γ, %) −→ (G, ρ) and (Zf−1, τf−1) : (G, ρ) −→ (Γ, %). Define the
denoted by τf. The map Zf−1 −→ Zf−1 defined by (x, γ) 7−→ (f (γ), f−1(x))
is clearly a G-Γ-equivariant Real homeomorphism; hence, (Zf−1, τf−1) and
(Zf−1, τf) are isomorphic Real generalized morphisms from (G, ρ) to (Γ, %).
Notice that Zf−1 is Zf as space; thus, (Zf, τf) is at the same time a Real
generalized morphism from (Γ, %) to (G, ρ) and from (G, ρ) to (Γ, %). Fur-thermore, it is simple to check that Zf ◦ Zf−1 and ZIdG define isomorphic
Real generalized morphisms from (G, ρ) into itself, and likewise, Zf−1◦ Zf and ZIdΓ are isomorphic Real generalized morphisms from (Γ, %) into itself.
Definition 1.24. Two Real groupoids (Γ, %) and (G, ρ) are said to be Morita equivalent if there exists a Real space (Z, τ ) that is at the same time a Real generalized morphism from Γ to G and from G to Γ; that is to say that Y oo r Z is a G-principal Real bundle and Z s //X is a Γ-principal Real bundle.
Remark 1.25. Given a Morita equivalence (Z, τ ) : (Γ, %) −→ (G, ρ), its inverse, denoted by (Z−1, τ ), is (Z, τ ) as Real space, and if [ : (Z, τ ) −→ (Z−1, τ ) is the identity map, the left RealG-action on (Z−1, τ ) is given by g · [(z) := [(z · g−1), and the right Real Γ-action is given by [(z) · γ := [(γ−1 · z); (Z−1, τ ) is the corresponding Real generalized morphism from (G, ρ) to (Γ, %).
The discussion before Definition 1.24 shows that the Real generalized mor-phism induced by a Real strict mormor-phism is actually a Morita equivalence. However, the converse is not true. Moreover, there is a functor
(1.2) RGs−→ RG,
where RGsis the category whose objects are Real locally compact groupoids
and whose morphisms are Real strict morphisms, given by f 7−→ Zf.
Definition 1.26 (Real cover groupoid). Let G ////X be a Real groupoid. Let U = {Uj} be a Real open cover of X. Consider the disjoint union
`
j∈JUj = {(j, x) ∈ J × X : x ∈ Uj} with the Real structure ρ(0) given by
ρ(0)(j, x) := (¯, ρ(x)) and define a Real local homeomorphism given by the projection π :`
jUj −→ X, (j, x) 7−→ x. Then the set
G[U] := {(j0, g, j1) ∈ J ×G × J : r(g) ∈ Uj0, s(g) ∈ Uj1},
endowed with the involution ρ(1)(j0, g, j1) := ( ¯j0, ρ(g), ¯j1) has a structure
of a Real locally compact groupoid whose unit space is `
jUj. The range
and source maps are defined by ˜r(j0, g, j1) := (j0, r(g)) and ˜s(j0, g, j1) :=
(j1, s(g)); two triples are composable if they are of the form (j0, g, j1) and
(j1, h, j2), where (g, h) ∈ G(2), and their product is given by (j0, g, j1) ·
It is a matter of simple verifications to check the following:
Lemma 1.27. Let G ////X be a Real groupoid, andU a Real open cover of X. Then the Real generalized morphism Zι :G[U] −→ G induced from the
canonical Real morphism
ι :G[U] −→ G, (j0, g, j1) 7−→ g,
is a Morita equivalence between (G[U], ρ) and (G, ρ). Definition 1.28. Let Z π s // X Y
be a locally trivialG-principal Real bundle. A section s : Y −→ Z is said to be Real if s ◦ % = τ ◦ s. Moreover, given a Real open cover {Uj}j∈J of Y , we
say that a family of local sections sj : Uj −→ Z is globally Real if for any
j ∈ J , we have
(1.3) s¯◦ % = τ ◦ sj.
Lemma 1.29. Any locally trivial G-principal Real bundle π : Z −→ Y admits a globally Real family of local sections {sj}j∈J over some Real open
cover {Uj}.
Proof. Choose a local trivialization (Ui, ϕi)i∈I of Z; i.e., ϕi : Ui −→ X are
continuous maps such that π−1(Ui) =: ZUi ∼= Ui×ϕi,X,rG with τZUi = (%, ρ).
It turns out that ZU(i,)
∼
= U(i,)×ϕ
i,X,rG, where
ϕi := ρ◦ ϕi◦ %: U
(i,)−→ X
is a well-defined continuous map and U(i,) := %(Ui) for (i, ) ∈ I × Z2.
However, for (i, ) ∈ I × Z2, there is a homeomorphism
U(i,)×ϕ i,X,rG
(%,ρ)//
U(i,)×ϕ+1 i ,X,r G .
Now, putting s(i,) : U(i,) −→ Z, x 7−→ (x, ϕi(x)), we obtain the desired
sections.
For the remainder of this subsection we will need the following construc-tion.
Let (Z, τ ) be a Real space and (Γ, %) a Real groupoid together with a continuous Real map ϕ : Z −→ Y . Then we define an induced groupoid ϕ∗Γ over Z in which the arrows from z1 to z2 are the arrows in Γ from ϕ(z1)
to ϕ(z2); i.e.,
ϕ∗Γ := Z ×ϕ,Y,rΓ ×s,Y,ϕZ ,
and the product is given by (z1, γ1, z2).(z2, γ2, z3) = (z1, γ1γ2, z3) whenever
γ1 and γ2 are composable, while the inverse is given by
Moreover, the triple (ρ, %, ρ) defines a Real structure ϕ∗% on ϕ∗Γ making it into a Real groupoid (ϕ∗Γ, ϕ∗%) that we will call the pull-back of Γ over Z via ϕ.
Lemma 1.30. Given a continuous locally split Real open map ϕ : Z −→ Y , then the Real groupoids Γ and ϕ∗Γ are Morita equivalent.
Proof. Consider the Real strict homomorphism ˜
ϕ : ϕ∗Γ 3 (z1, γ, z2) 7−→ γ ∈ Γ.
Then by Example 1.18 we obtain a Real generalized homomorphism
Z Zϕ˜
π1
oo s◦π2 //
Y
with Zϕ˜ := Z ×ϕ,Y,r˜ Γ, π1 and π2 the obvious projections, and where Z ,→
ϕ∗Γ by z 7−→ (z, ϕ(z), z). Now using the constructions of Example 1.18, it is very easy to check that Zϕ˜ is in fact a Morita equivalence.
Proposition 1.31. Two Real groupoids (Γ, %) and (G, ρ) are Morita equiv-alent if and only if there exist a Real space (Z, τ ) and two continuous Real maps ϕ : Z −→ Y and ϕ0 : Z −→ X such that ϕ∗Γ ∼= (ϕ0)∗G under a Real (strict) homeomorphism.
Proof. Let Y oo r Z s //X be a Morita equivalence. Let us define Γ n Z ∗ Z o G := {(γ, z1, z2, g) ∈ (Γ ×s,Y,rZ) × (Z ×s,X,r G) | z1g = γz2}.
This defines a Real groupoid over Z whose range and source maps are defined by the second and the third projection respectively, the product is given by
(γ, z1, z2, g) · (γ0, z2, z3, g0) = (γγ0, z1, z3, gg0),
provided that γ, γ0 ∈ Γ(2) and g, g0 ∈ G(2), and the inverse of (γ, z
1, z2, g)
is (γ−1, z2, z1, g−1). Now, for a given triple (z1, γ, z2) ∈ r∗Γ, the relations
r(z1) = r(γ) and r(z2) = s(γ) give r(γz2) = r(z1); then since r : Z −→ Y is
a RealG-principal bundle, there exists a unique g ∈ G such that γz2 = z1g.
This gives an injective homomorphism
Ψ : r∗Γ −→ Γ n Z ∗ Z o G, (z1, γ, z2) 7−→ (γ, z1, z2, g),
which respects the Real structures. In the other hand, the map Φ : Γ n Z ∗ Z o G −→ r∗Γ,
(γ, z1, z2, g) 7−→ (z1, γ, z2),
is a well-defined Real homomorphism that is injective and Real. Moreover, these two maps are, by construction, inverse to each other so that we have a Real homeomorphism r∗Γ ∼= Γ n Z ∗ Z o G. Furthermore, since s : Z −→ X is a Real Γ-principal bundle, we can use the same arguments to show that s∗G ∼= Γ n Z ∗ Z o G under a Real homeomorphism.
Conversely, if ϕ : Z −→ Y and ϕ0 : Z −→ X are given continuous Real maps and f : ϕ∗Γ −→ (ϕ0)∗X is a Real homeomorphism of groupoids, then the induced Real generalized homomorphism
ϕ∗Γ−→ (ϕZf 0)∗G
is a Morita equivalence and Lemma 1.30 completes the proof. The following example provides a characterization of groupoids Morita equivalent to a given Real space.
Example 1.32. Let (X, ρ), (Y, %) be a locally compact Hausdorff Real spa-ces, and let π : (Y, %) −→ (X, ρ) be a continuous locally split Real open map. Form the Real groupoid Y[2] ////Y , where Y[2] is the
fibered-product Y ×π,X,πY equipped with the obvious Real structure; the groupoid
structure on Y[2] is:
s(y1, y2) := y2; r(y1, y2) := y1;
(y1, y2)−1:= (y2, y1); (y1, y2) · (y2, y3) := (y1, y3).
Then the Real groupoids Y[2] ////Y and X ////X are Morita
equiv-alent. Indeed, we have π∗X ∼M orita X, thanks to Lemma 1.30; but π∗X
clearly identifies with Y[2] as Real groupoids.
Conversely, suppose (Γ, %) is a Real groupoids Morita equivalent to X. Then in view of Proposition 1.31, there is a Real space (Z, τ ), two continuous locally split Real open maps s : Z −→ X, r : Z −→ Y such that s∗X ∼= r∗Γ as Real groupoids over Z. In particular, r : Z −→ Y is a principal Real X-bundle, so that the Real space Y is homeomorphic to the quotient Real space Z/X = Z. Thus, we have isomorphism of Real spaces
r∗Γ = Z ×Y Γ ×Y Z ∼= Y ×Y Γ ×Y Y ∼= Γ.
Moreover, we have s∗X ∼= Z[2]as Real spaces. Therefore, the Real groupoids Γ ////Y and Z[2] ////Z as isomorphic.
Proposition 1.33 (Cf. Proposition 2.3 [25]). Any Real generalized mor-phism
Y oo r Z s //X
is obtained by composition of the canonical Morita equivalence between (Γ, %) and (Γ[U], %), where U is an open cover of Y , with a Real strict morphism fU: Γ[U] −→ G (i.e., its induced morphism in the category RG).
Proof. From Lemma 1.30, there is a Real Morita equivalence Z˜r: r∗Γ −→ Γ
and the Real homeomorphism r∗Γ ∼= Γ n Z ∗ Z o G induces a Real strict homomorphism f : r∗Γ −→ G given by the fourth projection, and hence a Real generalized homomorphism Zf : r∗Γ −→G. Furthermore, by using the
the composition Z˜r×ΓZ is r∗Γ-G-equivariently homeomorphic to Z (under
a Real homeomorphism); i.e., the diagram
Γ Z r∗Γ Z˜r ∼ = oo Zf G is commutative in the category RG.
Consider a Real open cover U = {Uj} of Y together with a globally
Real family of local sections sj : Uj −→ Z of r : Z −→ Y . Then, setting
(j0, γ, j1) 7−→ (sj0(r(γ)), γ, sj1(s(γ))) for (j0, γ, j1) ∈ Γ[U], we get a Real
strict homomorphism ˜s : Γ[U] −→ r∗Γ such that the composition Γ[U] −→ r∗Γ −→ Γ is the canonical map ι described in Example 1.26. Then, f ◦ ˜s : Γ[U] −→ G is the desired Real strict homomorphism. This proposition leads us to think of a Real generalized homomorphism from a Real groupoid (Γ, %) to a Real groupoid (G, ρ) as a Real strict mor-phism fU: (Γ[U], %) −→ (G, ρ), where U is a Real open cover of Y .
To refine this point of view, given two Real groupoids (Γ, %) and (G, ρ), let Ω denote the collection of such pairs (U, fU). We say that two pairs (U, fU) and (U0, fU0) are isomorphic provided that Zf
U◦ Zι−1U ∼= ZfU0 ◦ Z −1 ιU0,
where ιU : (Γ[U], %) −→ (Γ, %) and ιU0 : (Γ[U0], %) −→ (Γ, %) are the
canon-ical morphisms; this clearly defines an equivalence relation. We denote by Ω ((Γ, %), (G, ρ)) the set of isomorphism classes of elements of Ω.
Let (U, fU) : (Γ, %) −→ (G0.ρ0) be an equivalence class in Ω ((Γ, %), (G0, ρ0)) and let (V, fV) : (G0, ρ0) −→ (G, ρ) be an element in Ω ((G0, ρ0), (G, ρ)). Let ιG0 : G0[V] −→ G0 be the canonical morphism, and let Zι−1
G0 : (G
0, ρ0) −→
(G0[V], ρ0) be the inverse of ZιG0. Next, we apply Proposition 1.33 to the Real
generalized morphism Zι−1
G0 ◦ ZfU : Γ[U] −→ G
0[V] to get a Real open cover U0
of Y containingU and a Real strict morphism ϕU0 : (Γ[U0], %) −→ (G0[V], ρ0).
Then, we pose
(1.4) (V, fV) ◦ (U, fU) := (U0, fU0),
with fU0 = fV◦ ϕU0; thus we get an element of Ω ((Γ, %), (G, ρ)). It follows
that there exists a category RGΩ whose objects are Real groupoids, and in
which a morphism from (Γ, %) to (G, ρ) is a class (U, fU) in Ω ((Γ, %), (G, ρ)). Example 1.34. Any Real strict morphism f : (Γ, %) −→ (G, ρ) can be identified with the pair (Y, f ), by considering the trivial Real open cover Y consisting of one set, and by viewing the groupoid Γ as the cover groupoid Γ[Y ]. In particular, RGs is a subcategory of RGΩ.
Example 1.35. Suppose that (Z, τ ) : (Γ, %) −→ (G, ρ) is a Real general-ized morphism. Then, Proposition 1.33 provides a unique class (U, fU) ∈ Ω((Γ, %), (G, ρ)).
Remark 1.36. Note that a class (U, fU) ∈ Ω ((Γ, %), (G, ρ)) is an isomor-phism in RGΩ if there exists (V, fV) ∈ Ω ((G, ρ), (Γ, %)) such that
(1.5) ZfU◦ Zι−1U ◦ ZfV ∼= ZιV and ZfV ◦ Zι−1V ◦ ZfU∼= ZιU,
where ιU: (Γ[U], %) −→ (Γ, %) and ιV : (G[U], ρ) −→ (G, ρ) are the canonical morphisms.
Proposition 1.37. Define F : RG −→ RGΩ by
(1.6) F(Z, τ ) := (U, fU),
where, if (Z, τ ) : (Γ, %) −→ (G, ρ) is a class of Real generalized morphisms, (U, fU) is the class of pairs corresponding to (Z, τ ).
Then F is a functor; furthermore, F is an isomorphism of categories. Proof. Suppose that (Z, τ ) : (Γ, %) −→ (G0, ρ0), (Z0, τ0) : (G0, ρ0) −→ (G, ρ) are morphisms in RG. Let
F(Z0◦ Z, τ × τ0) = (U, fU) ∈ Ω ((Γ, %), (G, ρ)) , F(Z, τ ) = (U0
, fU0) ∈ Ω (Γ, %), (G0, ρ0) ,
F(Z0, τ0) = (V, fV) ∈ Ω (G0, ρ0), (G, ρ) .
Consider a Real open cover ˜U of Y containing U0 and a Real morphism ϕU˜ : (Γ[ ˜U], %) −→ (G0[V], ρ0) such that ZϕU˜ ◦ Z
−1 i ∼= Z
−1
ιV ◦ ZfU0 as Real
generalized morphisms from (Γ[U0], %) to (G0[V], ρ0), where
i : (Γ[ ˜U], %) −→ (Γ[U0], %) and ιV: (G0[V], ρ0) −→ (G0, ρ0)
are the canonical morphisms. Note that if ιU˜ : (Γ[ ˜U], %) −→ (Γ, %) is the
canonical morphism, then ιU˜ = ιU0◦ i; hence, Zι−1 ˜ U ∼ = Zi−1◦ Z−1 ιU0 by functori-ality.
On the other hand, F(Z0, τ0) ◦ F(Z, τ ) = (V, fV) ◦ (U, fU) = ( ˜U, fU˜), where
fU˜ = fV◦ ϕU˜. Henceforth, ZfU˜ ◦ Z −1 ιU˜ ∼ = ZfV◦ Zϕ˜U◦ Zi−1◦ Zι−1U0 ∼= ZfV ◦ Zι−1V ◦ ZfU0 ◦ Zι−1U0 ∼= Z 0◦ Z,
which shows that F(Z0 ◦ Z, τ × τ0) ∼= F(Z0, τ0) ◦ F(Z, τ ), and thus F is a
functor.
Now, it is not hard to see that we get an inverse functor for F by defining (1.7) Z : RGΩ−→ RG, (U, fU) 7−→ (ZfU◦ Zι−1U , τ ),
where τ is defined in an obvious way.
1.5. Real graded twists. In this section we define Real graded twists. Definition 1.38 (Cf. [11, §2]). Let Γ ////Y be a Real groupoid and let S be a Real Abelian group. A Real graded S-twist (eΓ, δ) over Γ consists of the following data:
(i) a Real groupoid eΓ whose unit space is Y , together with a Real strict homomorphism π : eΓ −→ Γ that restricts to the identity in Y ,
(ii) a (left) Real action of S on eΓ compatible with the partial product in
e
Γ making eΓ
π //
Γ a (left) Real S-principal bundle,
(iii) a strict homomorphism δ : Γ −→ Z2, called the grading, such that
δ(¯γ) = δ(γ) for any γ ∈ Γ.
In this case we refer to the triple (eΓ, Γ, δ) as a Real graded S-twist, and it is sometimes symbolized by the “extension”
S //Γe π // Γ δ Z2
Example 1.39 (The trivial twist). Given Real groupoid Γ, we form the product groupoid Γ × S and we endow it with the Real structure (γ, λ) := (¯γ, ¯λ) for. Let S act on Γ × S by multiplication with the second factor. Then T0:= (Γ × S, 0) is a Real graded twist of Γ, where 0 : Z2 −→ Z2 is the zero
map. This element is called the trivial Real graded S-twist over Γ.
Example 1.40. Let Y be a locally compact Real space and {Ui}i∈I×{±1}be
a good Real open. Let us consider the Real groupoid Y [U] ////`
iUi , and
the space Y × S together with the Real structure (y, λ) 7−→ (¯y, ¯λ) and the Real S-action given by the multiplication on the second factor. We write xi0i1
for (i1, x, i1) ∈ Y [U]. There is a canonical Real morphism δ : Y [U] −→ Z2
given by δ(xi0i1) := ε0+ ε1 for i0 = (i00, ε0), i1 = (i01, ε1) ∈ I. Then, a Real
graded S-twist (eΓ, Y [U], δ) consists of a family of principal Real S-bundles e
Γij ∼= Uij × S subject to the multiplication
(xi0i1, λ1) · (xi1i2, λ2) = (xi0i2, λ1λ2ci0i1i2(x)),
where c = {ci0i1i2} is a family of continuous maps ci0i1i2 : Ui0i1i2 −→ S
which is a 2-cocycle such that c¯i0¯i1¯i2(¯x) = ci0i1i2(x) for all x ∈ Ui0i1i2 =
Ui0 ∩ Ui1 ∩ Ui2. The pair (δ, c) will be called the Dixmier–Douady class of
(eΓ, Y [U], δ) (see Section 2.12).
Example 1.41. Let Γ ////Y be a Real groupoid, and let J : Λ −→ Y be a Real S-principal bundle. Then the tensor product r∗Λ ⊗ s∗Λ, which is a Real S-principal bundle over Γ, naturally admits the structure of Real groupoid over Y , so that (r∗Λ ⊗ s∗Λ, 0) is a Real graded S-twist over Γ.
There is an obvious notion of strict morphism of Real graded S-twists. For instance, two Real graded S-twists (eΓ1, Γ, δ1) and (eΓ2, Γ, δ2) are isomorphic
such that the diagram e Γ1 π1 // f Γ e Γ2 π2 @@
commutes in the category RGs. In particular, we say that (eΓ, δ) is strictly
trivial if it isomorphic to the trivial Real graded groupoid (Γ × S, 0). By [
TwR(Γ, S) we denote the set of strict isomorphism classes of Real graded S-twists over Γ. The class of (eΓ, δ) in [TwR(Γ, S) is denoted by [eΓ, δ]. Definition 1.42 (Cf. [11, 23, 6]). Given two Real graded S-twists T1 =
(eΓ1, δ1) and T2= (eΓ2, δ2) over G, we define their tensor product
T1⊗ˆT2 = (eΓ1⊗eˆΓ2, δ1+ δ2)
by the Baer sum ofT1 andT2 defined as follows. Define the groupoid eΓ1⊗eˆΓ2
as the quotient
(1.8) Γe1×ΓΓe2/S := {( ˜γ1, ˜γ2) ∈ eΓ1×π1,Γ,π2 eΓ2}/( ˜γ1, ˜γ2)∼(λ ˜γ1,λ−1γ2)˜ ,
where λ ∈ S, together with the obvious Real structure. The projection π1⊗ π2 is just πi and δ = δ1+ δ2 is given by δ(γ) = δ1(γ) + δ2(γ).
The product in the Real groupoid eΓ1⊗eˆΓ2 is
(1.9) ( ˜γ1, ˜γ2)( ˜γ01, ˜γ20) := (−1)δ2(γ2)δ1(γ 0 1)( ˜γ1γ˜0
1, ˜γ2γ˜20),
whenever this does make sense and where γi = π2( ˜γi), i = 1, 2.
Lemma 1.43 ([23, p.4]). Given [eΓi, δi] ∈ [TwR(Γ, S), i = 1, 2, set
[eΓ1, δ1] + [eΓ2, δ2] := [eΓ1⊗eˆΓ2, δ1+ δ2].
Then, under this sum, [TwR(Γ, S) is an Abelian group whose zero element is given by the class of the trivial elementT0 = (G × S, 0).
Proof. The tensor product defined above is commutative in [TwR(Γ, S). Indeed, the groupoid eΓ2⊗eˆΓ1 = eΓ2×ΓΓe1/S is endowed with the multiplication
( ˜γ2, ˜γ1)( ˜γ20, ˜γ10) = (−1)δ1(γ1)δ2(γ 0 2)( ˜γ2γ˜0
2, ˜γ1γ˜10).
Then the map
e
Γ1⊗eˆΓ2 −→ eΓ2⊗eˆΓ1 , ( ˜γ1, ˜γ2) 7−→ (−1)δ1(γ1)δ2(γ2)( ˜γ2, ˜γ1)
is a Real S-equivariant isomorphism of groupoids.
Now define the inverse of (eΓ, δ) is (eΓop, δ) where eΓop is eΓ as a set but,
together with the same Real structure, but the S-principal bundle structure is replaced by the conjugate one, i.e., λ˜γop= (¯λ˜γ)op, and the product ∗
opin e Γop is ˜ γ ∗opγ˜0 := (−1)δ(γ)δ(γ 0) ˜ γ ˜γ0.
Now it is easy to see that the map
Γ × S −→ eΓ ×ΓeΓop/S , (γ, λ) 7−→ (λ˜γ, ˜γ) ,
where ˜γ ∈ eΓ is any lift of γ ∈ Γ, is an isomorphism. We have the following criteria of strict triviality; the proof is the same as in [25, Proposition 2.8].
Proposition 1.44. Let (eΓ, δ) be a Real graded S-twist over the Real groupoid Γ ////Y . The following are equivalent:
(i) (eΓ, δ) is strictly trivial.
(ii) δ(γ) = 0, ∀γ ∈ Γ, and there exists a Real strict homomorphism σ : Γ −→ eΓ such that π ◦ σ = Id.
(iii) δ(γ) = 0, ∀γ ∈ Γ,, and there exists a Real S-equivariant groupoid homomorphism ϕ : eΓ −→ S.
Example 1.45. Let J : Λ −→ Y be a Real S-principal bundle with a Real (left) Γ-action that is compatible with the S-action; in other words
Y Λ
J
oo //? is a Real generalized homomorphism from Γ to S. Then,
the Real Γ-action induces an S-equivariant isomorphism Λs(γ) 3 v 7−→ γ · v ∈ Λr(γ) for every γ ∈ Γ. Hence, there is a Real S-equivariant groupoid
isomorphism ϕ : r∗Λ ⊗ s∗Λ −→ Γ × S defined as follows. If (v, [(w)) ∈
Λr(γ)⊗ Λs(γ), there exists a unique λ ∈ S such that γ · w = v · λ. We then
set
ϕ([v, [(w)]) := (γ, λ).
The inverse of ϕ is ϕ0(γ, λ) := [vγ, γ−1· vγ], where for γ ∈ Γ, vγ is any lift
of r(γ) through the projection J .
Observe that the set of Real graded S-twists of the from (r∗Λ ⊗ s∗Λ, 0) over Γ (see Example 1.41) is a subgroup of [TwR(Γ, S). By [extR(Γ, S) we denote the quotient of [TwR(Γ, S) by this subgroup.
Let us show that [extR(·, S) is functorial in the category RGs. Let Γ, Γ0
be two Real groupoids, and let f : Γ0 −→ Γ be a morphism in RGs. Suppose thatT = (eΓ, δ) is a Real graded S-twist over Γ. Then, the pull-back
f∗Γ := ee Γ ×π,Γ,fΓ0
of the Real S-principal bundle π : eΓ −→ Γ, on which the Real groupoid structure is the one induced from the product Real groupoid eΓ × Γ0, defines a Real graded twist
(1.10) f∗T := S //f∗eΓ f∗π //Γ0 f∗δ Z2
where f∗π(˜γ, γ0) := γ0, f∗δ(γ0) := δ(f (γ0)) ∈ Z2, and the Real left
S-action on f∗Γ being given by λ · (˜e γ, γ0) = (λ˜γ, γ0). Suppose now that Ti= (eΓi, δi), i = 1, 2 are representatives in [extR(Γ, S). Then,
f∗(T1⊗ˆT2) = f∗T1⊗fˆ ∗T2; indeed, f∗(eΓ1⊗eˆΓ2) = e Γ1×ΓΓe2/S ×ΓΓ0∼= (Γ1×ΓΓ0) ×Γ(eΓ2×ΓΓ0) /S = f∗eΓ1⊗fˆ ∗Γe2.
Moreover, it is easily seen that if T1 and T2 are equivalent in [extR(Γ, S),
then so are f∗T1 and f∗T2. Thus, f induces a morphism of Abelian groups
f∗: [extR(Γ, S) −→ [extR(Γ0, S). We then have proved this: Lemma 1.46. The correspondence
[
extR(·, S) : RGs−→ Ab,
(1.11)
Γ 7−→ [extR(Γ, S), f 7−→ f∗,
where Ab is the category of Abelian groups, is a contravariant functor. In particular, [extR(G, S) is invariant under Real strict isomorphisms.
1.6. Real graded central extensions. In this subsection we introduce Real graded central extensions of Real groupoids, by adapting [11, 12, 6, 23] to our context.
Definition 1.47. Let (eΓi, Γi, δi), i = 1, 2, be Real graded S-twists. Then a
Real generalized homomorphism Z : eΓ1 −→ eΓ2 is said to be S-equivariant if
there is a Real action of S on Z such that
(λ ˜γ1) · z · ˜γ2 = ˜γ1· (λz) · ˜γ2 = ˜γ1· z · (λ ˜γ2),
for any (λ, ˜γ1, z, ˜γ2) ∈ S × eΓ1 × Z × eΓ2 such that these products make
sense. We refer to Z : (eΓ1, Γ1, δ1) −→ (eΓ2, Γ2, δ2) as a generalized morphism
of Real graded S-twists. In particular, if Z is an isomorphism, the two Real graded S-twists are said to be Morita equivalent ; in this case we write (eΓ1, Γ1, δ1) ∼ (eΓ2, Γ2, δ2).
Lemma 1.48. Let Z : (eΓ1, Γ1, δ1) −→ (eΓ2, Γ2, δ2) be a generalized
mor-phism. Then the S-action on Z is free and the Real space Z/S (with the obvious involution) is a Real generalized homomorphism from Γ1 to Γ2.
Proof. Same as [25, Lemma 2.10].
Definition 1.49. Let G be a Real groupoid and S an abelian Real group. A Real graded S-central extension ofG consists of a triple (Γ, Γ, δ, P ), wheree (eΓ, Γ, δ) is a Real graded S-twist, and P is a (Real) Morita equivalence Γ −→G.
Definition 1.50. We say that (eΓ1, Γ1, δ1, P1) and (eΓ2, Γ2, δ2, P2) are Morita
equivalent if there exists a Morita equivalence Z : (eΓ1, Γ1, δ1) −→ (eΓ2, Γ2, δ2)
such that the diagrams
(1.12) Γ1 Z/S // P1 Γ2 P2 G and (1.13) Γ1 Z/S // δ1 Γ2 δ2 Z2
commute in the category RG. Such a Z is also called an equivalence bimodule of Real graded S-central extensions. The set of Morita equivalence classes of Real graded S-central extensions ofG is denoted by [ExtR(G, S).
The set [ExtR(G, S) admits a natural structure of abelian group described in the following way. Assume that Ei = (eΓi, Γi, δi, Pi), i = 1, 2, are two
given Real graded S-central extensions of G, then Y1 Z r
oo s //
Y2 is
a Morita equivalence between Γ1 and Γ2, where Z = P1×GP2. But from
Proposition 1.31 there exists a Real homeomorphism f : s∗Γ2 −→ r∗Γ1. Now
one can see that the maps π : r∗eΓ1 −→ r∗Γ1, (z, ˜γ1, z0) 7−→ (z, π1( ˜γ1), z0) and π0 : s∗eΓ2 −→ r∗Γ1(z, ˜γ2, z0) 7−→ π ◦ f (z, ˜γ2, z0) define two Real S-principal bundles and then (r∗eΓ1, δ) and (s∗Γe2, δ), where δ := δ1◦ pr2, define elements of [extR(r∗Γ1, S). Therefore, we can form the tensor product (r∗Γe1⊗sˆ ∗Γe2, δ ⊗ δ) are Real graded S-groupoid over r∗Γ1. Moreover, r∗Γ1 ∼M orita Γ1; then,
if P : r∗Γ1 −→ G is a Real Morita equivalence, we obtain a Real graded
S-central extension of G by setting
(1.14) E1⊗Eˆ 2:= (r∗Γe1⊗sˆ ∗Γe2, r∗Γ1, δ, P ),
that we will call the tensor product of E1 and E2. Thus, we define the sum
[E1] + [E2] := [E1⊗Eˆ 2],
which is easily seen to be well-defined in [ExtR(G, S). The inverse Eopof E is (eΓop, Γ, δ, P ). Notice that [extR(G, S) is naturally a subgroup of [ExtR(G, S) by identifying a Real graded S-twist (eΓ,G, δ) with the Real graded S-central extension (eΓ,G, δ, G). We summarize this in the next lemma.
Lemma 1.51. Under the sum defined above, [ExtR(G, S) is an abelian group whose zero element is the class of the trivial Real graded S-central extension (G × S, G, 0, G).
When the Real structure is trivial, then we recover the usual definition of graded central extensions (see [6] for instance) ofG by the group Z2.
Proposition 1.52. Suppose that G ////X is equipped with a trivial Real structure. Then
[
ExtR(G, S1) ∼= dExt(G, Z2).
Example 1.53. Suppose G reduces to a Real space X. Then following Example 1.32, a Real graded S-central extension of X is a triple (eΓ, Y[2], δ), where Y is a Real space together with a continuous locally split Real open map π : Y −→ X, and δ : Y[2] −→ Z2 is a Real morphism.
In particular, suppose ρ is trivial. Then, by Proposition 1.52, giving a Real graded S1-central extension of X amounts to giving a real bundle gerbe
Z2 //eΓ Y[2] ////Y π X
in the sense of Mathai, Murray, and Stevenson [14], together with an aug-mentation δ : Y[2] −→ Z2.
1.7. Functoriality of \ExtR(·, S). The aim of this subsection is to show that [ExtR(·, S) is functorial in the category RG, and hence that the group
[
ExtR(G, S) invariant under Morita equivalence. To do this, we will need the following:
Proposition 1.54. Let G ////X be a Real groupoid. Then, there is an isomorphism of abelian groups
(1.15) ExtR([ G, S) ∼= lim−→
U
[
extR(G[U], S).
Before giving the proof of this proposition, we have to describe the sum in the inductive limit
lim −→
U
[
extR(G[U], S).
LetU1 and U2 be two Real open covers of X, and let Ti = (˜Gi,G[Ui], δi) be
Real graded S-groupoids over G[Ui], i = 1, 2. Let (V, fV) ∈ Ω (G[U1],G[U2])
be the unique class corresponding to the Real Morita equivalence Zι−1U1◦ ZιU2
from G[U1] toG[U2]. V is a Real open cover of X containing U1, and
is a Real strict morphism. Denote by ιV,U1 the canonical Real morphism G[V] −→ G[U1]. Then, the tensor product ofT1 and T2 is
(1.16) T1⊗ˆT2:= ιV,U1∗ T1⊗fˆ V∗T2,
which defines a Real graded S-groupoids over the Real groupoidG[V]. Proof of Proposition 1.54. For a Real graded S-central extension E = (eΓ, Γ, δ, P ) of G , let (V, fV) ∈ Ω (G, Γ) be the isomorphism in RGΩ
corre-sponding to the Morita equivalence P−1 :G −→ Γ. Setting (1.17) TE:= S //fV∗Γe f∗ Vπ // G[V] δ◦fV Z2
we get a Real graded S-groupoid overG[V]. It is not hard to check that this provides us the desired isomorphism of abelian groups; the inverse is given by the formula
(1.18) ET := (˜G, G[U], δ, ZιU),
for a Real graded S-twistT = (˜G, G[U], δ).
From this proposition, it is now possible to define the pull-back of a Real graded S-central extension via a Real generalized morphism. More precisely, we have
Definition and Proposition 1.55. Let G and G0 be Real groupoids, and let Z : G0 −→ G be a Real generalized morphism. Let E = (eΓ, Γ, δ, P ) is be a representative in [ExtR(G, S), and TE = (fV∗Γ,e G[V], δ ◦ fV) its image in lim
−→
U
[
extR(G[U], S) (see the proof of Proposition 1.54). Let (W, fW) ∈ Ω G0,G[V]
be the morphism in RGΩ corresponding to the Real generalized morphism
Zι−1V ◦ Z :G0−→G[V]. Then
(1.19) Z∗E := Ef∗
WTE.
is a Real graded S-central extension of the Real groupoidG0; it is called the pull-back of E along Z
Now the following is straightforward.
Corollary 1.56. There is a contravariant functor
(1.20) ExtR(·, S) : RG −→ Ab,[
which sends a Real groupoidG to the abelian group [ExtR(G, S). In particular, [
2. Real ˇCech cohomology
2.1. Real simplicial spaces. We start by recalling some preliminary no-tions. For each zero integer n ∈ N, we set [n] = {0, . . . , n}. Recall [21] that the simplicial (resp. pre-simplicial) category ∆ (resp. ∆0) is the category whose objects are the sets [n], and whose morphisms are the nondecreasing (resp. increasing) maps f : [m] −→ [n]. For n ∈ N, we denote by ∆(N ) the N -truncated full subcategory of ∆ whose objects are those [k] with k ≤ N . Definition 2.1. A Real simplicial (resp. pre-simplicial, N -simplicial) topo-logical space consists of a contravariant functor from ∆ (resp. ∆0, ∆(N )) to the category RTop whose objects are topological Real spaces and mor-phisms are continuous Real maps. A morphism of Real simplicial (resp. pre-simplicial, . . . ) spaces is a morphism of such functors.
More concretely, a Real (pre-)simplicial space is given by a family (X•, ρ•) = (Xn, ρn)n∈N
of topological Real spaces, and for every map f : [m] −→ [n] we are given a continuous Real map (called face or degeneracy map depending which of m and n is larger) ˜f : (Xn, ρn) −→ (Xm, ρm) , satisfying the relation
]
f ◦ g = ˜g ◦ ˜f whenever f and g are composable.
Definition 2.2. Let (X•, ρ•) be a Real simplicial space. For any N ∈ N, the
N -skeleton of (X•, ρ•) is the Real simplicial space (X•, ρ•)N “of dimension
N ”; that is, (Xn, ρn)N = (Xn, ρn) for n ≤ N , and (Xn, ρn)N = (XN, ρN)
for all n ≥ N + 1.
Let εni : [n − 1] −→ [n] be the unique increasing injective map that avoids i, and let ηn
i : [n + 1] −→ [n] be the unique nondecreasing surjective map
such that i is reached twice; that is,
εni(k) = ( k, if k ≤ i − 1, k + 1, if k ≥ i, (2.1) ηin(k) = ( k, if k ≤ i; k − 1, if k ≥ i + 1. We will omit the superscript n if there is no ambiguity.
If (X•, ρ•) is a Real simplicial space, it is straightforward to check that
the face and degeneracy maps ˜
εni : (Xn, ρn) −→ (Xn−1, ρn−1),
˜
ηni : (Xn, ρn) −→ (Xn+1, ρn+1),
i = 0, . . . , n satisfy the following simplicial identities: ˜
εn−1i ε˜nj = ˜εn−1j−1ε˜ni if i ≤ j − 1, (2.2)
˜
˜ εn+1i η˜jn= ˜ηj−1n−1ε˜ni if i ≤ j − 1, ˜ εn+1i η˜jn= ˜ηjn−1ε˜ni−1 if i ≥ j + 2, ˜ εn+1j η˜jn= ˜εn+1j+1η˜jn= IdXn.
Conversely, let (Xn, ρn)n∈N be a sequence of topological Real spaces
to-gether with maps satisfying (2.2). Then thanks to [13, Theorem 5.2], there is a unique Real simplicial structure on (X•, ρ•) such that ˜εi and ˜ηi are the
face and degeneracy maps respectively.
Example 2.3 (Cf. [24, §2.3]). Consider the pair groupoid
[n] × [n] ////[n];
that is, the product is (i, j)(j, k)) := (i, k) and the inverse of (i, j) is (j, i). If (G, ρ) is a topological Real groupoid, we define
Gn:= Hom([n] × [n],G)
as the space of strict morphisms from the groupoid [n] × [n] ////[n] to G ////X . We obtain a Real structure on Gn by defining ρn(ϕ) := ρ ◦ ϕ,
for ϕ ∈ Gn. Any f ∈ Hom∆([m], [n]) (or f ∈ Hom∆0([m], [n])) naturally
gives rise to a strict morphism f × f : [m] × [m] −→ [n] × [n], which, in turn, induces a Real map ˜f : (Gn, ρn) −→ (Gm, ρm) given by ˜f (ϕ) := ϕ ◦ (f × f )
for ϕ ∈Gn. Hence, we obtain a Real simplicial space (G•, ρ•).
Notice that the groupoid
[n] × [n] ////[n]
is generated by elements (i−1, i), 1 ≤ i ≤ n; indeed, given an element (i, j) ∈ [n] × [n], we can suppose that i ≤ j (otherwise, we take its inverse (j, i)), and then (i, j) = (i, i + 1) . . . (j − 1, j). It turns out that any strict morphism ϕ : [n]×[n] −→G is uniquely determined by its images ϕ(i−1, i) ∈ G; hence, the well-defined Real map
Gn−→G(n), ϕ 7−→ (g1, . . . , gn),
where gi := ϕ(i − 1, i), 1 ≤ i ≤ n, and
G(n):= {(h
1, . . . , hn) | s(hi) = r(hi−1), i = 1, . . . , n},
identifies (Gn, ρn) with (G(n), ρ(n)), where ρ(n) is the obvious Real structure
on the fibred product G(n). Therefore, using this identification, the face
maps ˜εni : (Gn, ρn) −→ (Gn−1, ρn−1) of (G•, ρ•) are given by:
˜ εn0(g1, g2, . . . , gn) = (g2, . . . , gn), (2.3) ˜ εni(g1, g2, . . . , gn) = (g1, . . . , gigi+1, . . . , gn), 1 ≤ i ≤ n − 1, ˜ εnn(g1, g2, . . . , gn) = (g1, . . . , gn−1),
and for n = 1, by ˜ε10(g) = s(g), ˜ε11(g) = r(g); while the degeneracy maps ˜
ηin: (Gn, ρn) −→ (Gn+1, ρn+1) are given by:
˜
ηn0(g1, g2, . . . , gn) = (r(g1), g1, . . . , gn),
(2.4)
˜
ηni(g1, g2, . . . , gn) = (g1, . . . , s(gi), gi+1, . . . , gn), 1 ≤ i ≤ n,
and ˜η00:G0 −→G1 is the unit map of the Real groupoid.
Now for n ∈ N, we define the space (EG)n of (n + 1)-tuples of elements
of G that map to the same unit; i.e.,
(EG)n:= {(γ0, . . . , γn) ∈Gn+1 | r(γ0) = r(γ1) = · · · = r(γn)}.
Suppose we are given (g1, . . . , gn) ∈ Gn. Then we can choose an (n +
1)-tuple (γ0, . . . , γn) ∈ (EG)n such that gi = γi−1−1γi for each i = 1, . . . , n. If
(γ00, . . . γ0n) is another (n + 1)-tuple satisfying these identities, then s(γi0) = s((γi−10 )−1γi0) = s(γi−1−1γi) = s(γi),
for all i = 1, . . . , n, and that means that there exists a unique g ∈ G, such that s(g) = r(γi) and γi0 = g · γi. This hence gives us a well-defined injective
map
Gn−→ (EG)n/∼, (g1, . . . , gn) 7−→ [γ0, . . . , γn],
where (γ0, . . . , γn) ∼ (g · γ0, . . . , g · γn). Moreover, this map is surjective,
for if (γ0, . . . , γn) ∈ (EG)n, one can consider morphisms gi from s(γi) to
s(γi−1), i = 1, . . . , n, so that we have
γ1 = γ0g1, γ2= γ1g2 = γ0g1g2, . . . , γn= γ0g1· · · gn,
and then
[γ0, . . . , γn] = [r(g1), g1, g1g2, . . . , g1· · · gn]
which gives the inverse (EG)n/∼ 3 [γ0, . . . , γn] 7−→ (g1, . . . , gn) ∈ Gn. It
hence turns out that we can identifyGnwith the quotient (EG)n. Note that
the quotient space (EG)n/∼ naturally inherits the Real structure ρn+1 and
that the isomorphism defined above is compatible with the Real structures. Henceforth, an element ofGn will be represented by a vector
− →g = (g
1, . . . , gn),
where we view −→g as a morphism [n] × [n] −→ G, and gi = −→g (i − 1, i),
i = 1, . . . , n, or −→g = [γ0, . . . , γn] as a class in (EG)n/∼. For the first picture,
if f ∈ Hom∆([m], [n]), then the Real face/degeneracy map ˜f : (Gn, ρn) −→
(Gm, ρm) is given by:
(2.5) f (−˜→g ) = (−→g (f (0), f (1)) , . . . , −→g (f (m − 1), f (m))) . For instance, if f in injective, then
−
→g (f (i − 1), f (i)) = −→g (f (i − 1), f (i − 1) + 1) · · · −→g (f (i) − 1, f (i))
for f (i) ≥ 1, and thus
However, the second picture offers a more general formula for the face and degeneracy maps; roughly speaking, for any f ∈ Hom∆([m], [n]), we have
−
→g (i, j) = γ−1
i γj for every (i, j) ∈ [n] × [n]. In particular,
−
→g (f (k − 1), f (k)) = γ−1
f (k−1)γf (k),
for every k ∈ [m]; then (2.5) gives:
(2.7) f (−˜→g ) = [γf (0), . . . , γf (m)].
2.2. Real sheaves on Real simplicial spaces. In this subsection we closely follow [21, §3] to study Real sheaves on Real (pre-)simplicial spaces. We start by introducing some preliminary notions.
LetC be a topological category. We define the category CR by setting:
• Ob(CR) consists of triples (A, σA, A0), where A, A0 ∈ Ob(C) and
σA∈ HomC(A, A0);
• HomCR((A, σA, A0), (B, σB, B0)) consists of pairs (f, ˜f ) of morphisms
f : A −→ B, ˜f : A0 −→ B0 inC such that the diagrams A σA f // B σB A0 ˜ f // B0 commute.
Now, let φ : C −→ C be a functor. Then we define the subcategory Cφ
of CR whose objects are pairs (A, φ(A)), where A ∈ Ob(C), and in which
a morphism from (A, φ(A)) to (B, φ(B)) is a pair (f, ˜f ) of morphisms f : A −→ B, ˜f : φ(A) −→ φ(B) such that ˜f ◦φ = φ◦f . A fundamental example of this is the category OB(X) of open subsets of a given topological Real space (X, ρ). Recall that objects of this category are the collection of the open sets U ⊂ X, and morphisms are the canonical injections V ,→ U when V ⊂ U . Given such a Real space (X, ρ), the map ρ induces a functor (which is an isomorphism) ρ : OB(X) −→ OB(X) given by
V ι //U 7−→ ρ(V ) ρ◦ι◦ρ//ρ(U ) .
Definition 2.4 (Real presheaves). Let (X, ρ) be a topological Real space, and let C be a topological category. A Real presheaf (F, σ) on (X, ρ) with values in C is a contravariant functor from OB(X)ρ to CR; a morphism of
Real presheaves is a morphism of such functors.
Specifically, from the fact that ρ : X −→ X is a homeomorphism and from the canonical properties of the injections V ,→ U of open sets V ⊂ U ⊂ X, a Real presheaf on (X, ρ) with values inC assigns to each open subset U ⊂ X a triple (F(U ), σU, F(ρ(U ))), where F(U ), F(ρ(U )) are objects of C, and
ϕV,U : F(U ) −→ F(V ) and ϕρ(V ),ρ(U ) : F(ρ(U )) −→ F(ρ(V )), called the restriction morphisms, such that:
• ϕU,U = IdF(U ).
• σV ◦ ϕV,U = ϕρ(V ),ρ(U ) ◦ σU.
• ϕW,U = ϕW,V ◦ ϕV,U, and ϕρ(W ),ρ(U ) = ϕρ(W ),ρ(V )◦ ϕρ(V ),ρ(U ).
A morphism of Real presheaves φ : (F, σF) −→ (G, σG) is then a family of φU ∈ HomC(F(U ), G(U )) such that, for all pairs of open sets U, V with V ⊂ U , the diagrams below commute:
(2.8) F(ρ(U )) φ ρ(U ) F(U ) σF U oo φU ϕF V,U // F(V ) φV G(ρ(U )) G(U ) σG U oo ϕ G V,U // G(V ).
As in the standard case, if (F, σ) is a Real presheaf over X, and if U is an open subset of X, an element s ∈ F(U ) is called a section of (F, σ) on U , and for x ∈ X. If V is an open subset of U , and s ∈ F(U ), one often writes s|V for ϕV,U(s).
Definition 2.5 ([10, Definition 2.2]). A Real sheaf over (X, ρ) with values inC is a Real presheaf (F, σ) satisfying the following conditions:
(i) For any open set U ⊂ X, any open cover U =S
i∈IUi, any section
s ∈ F(U ), s|Ui = 0 for all i implies s = 0.
(ii) For any open set U ⊂ X, any open cover U = S
i∈IUi, any
fam-ily of sections si ∈ F(Ui) satisfying si|Uij = sj |Uij for all nonempty
intersection Uij, there exists s ∈ F(U ) such that s|Ui = si for all i.
A morphism of Real sheaves is a morphism of the underlying presheaves. We denote byCR(X) (or simply by Shρ(X) if there is no risk of confusion)
for the category of Real sheaves on (X, ρ) with values in C.
Notice that if (F, σ) is a Real sheaf (resp. presheaf) on (X, ρ), then F is a sheaf (resp. presheaf) on X in the usual sense. Recall that the stalk of F at a point x ∈ X, denoted by Fx, is the direct limit of the direct system
(F(U ), ϕV,U) where U runs along the family of open neighborhoods of x; i.e., Fx:= lim−→
x∈U
F(U ),
The image of a section s ∈ F(U ) in Fx by the canonical morphism
F(U ) −→ Fx
(where x ∈ U ) is called the germ of s at x and denoted by sx.
Note that if U is an open neighborhood of x, ρ(U ) is an open neighborhood of ρ(x), and the isomorphism σU : F(U ) 3 s 7−→ σU(s) ∈ F(ρ(U )) extends
inverse is σρ(x). We thus have a well-defined 2-periodic isomorphism, also denoted by σ, on the topological 2space F := `x∈XFx, given by
(2.9) σ :F −→ F, (x, sx) 7−→ (ρ(x), σx(sx))
which gives a Real space (F, σ).
Example 2.6. Let (X, ρ) be a Real space. Then the space C (X) of continu-ous complex values functions on X defines a Real sheaf of abelian groups on (X, ρ) by (U, ρ(U )) 7−→ (C (U ), ˜ρU, C (ρ(U ))), where ˜ρU(f )(ρ(x)) := f (x). Definition 2.7 (Pushforward, pullback). Let (X, ρ), (Y, %) be topological Real spaces, f : (Y, %) −→ (X, ρ) a continuous Real map. Suppose that (F, σ) and (G, ς) are Real sheaves on (X, ρ) and (Y, %) respectively, with values in the same categoryC.
(i) The pushforward of (G, ς) by f , denoted by (f∗G, f∗ς), is the Real
sheaf on (X, %) defined by the contravariant functor:
(2.10) OB(X)ρ−→CR, (U, ρ(U )) 7−→ (f∗G(U ), f∗ςU, f∗G(ρ(U ))) ,
where f∗G(U ) := G(f−1(U )), f∗ςU := ςf −1(U ), and
f∗G(ρ(U )) = G(f−1(ρ(U ))) ∼= G(%(f−1(U ))).
(ii) The pullback of (F, σ) along f , denoted by (f∗F, f∗σ), is the Real sheaf on (Y, %) associated to the Real presheaf defined by:
(2.11) OB(Y )%−→CR, (V, %(V )) 7−→ (f∗F(V ), f∗σV, f ∗ F(%(V ))), where f∗F(V ) := lim−→ f (V )⊂U ⊂X U open F(U ), and f∗σV : f ∗F(V ) −→ f∗F(%(V ))
is the morphism inC extending functorially σU : F(U ) −→ F(ρ(U )) along the family of open neighborhoods of f (V ) in X.
It immediately follows from this definition that we have a covariant func-tor RTop−→ RSh, (2.12) (Y, %) f //(X, ρ) 7−→ Sh%(Y ) f∗ // Shρ(X) ,
and a contravariant functor
RTop−→ RSh, (2.13) (Y, %) f //(X, ρ) 7−→ Shρ(X) f∗ //Sh%(Y ),
2Recall that if F is a presheaf over X, any section s ∈ F(U ) induces a map [s] :
U −→`
xFx, y 7−→ sy. We give F := `x∈XFx the largest topology such that all the
maps [s] are continuous. On the other hand, associated to F, there is a sheaf bFgiven by b
F(U ) := Γ(U,F), and we have that F(U) ∼= Γ(U,F) if and only if F is a sheaf. Then, given a Real presheaf (F, σ), one can define its associated Real sheaf in the same fashion.
where RSh is the category whose objects are the categories of Real sheaves on given Real spaces and morphisms are functors of such categories.
We will also need the following proposition.
Proposition 2.8. Let f : (Y, %) −→ (X, ρ) be a a continuous Real map. Suppose that (F, σ) and (G, ς) are Real sheaves on (X, ρ) and on (Y, %) re-spectively, with values in the same category C. Then
(2.14) HomShρ(X)((F, σ), (f∗G, f∗ς)) ∼= HomSh%(Y )((f∗F, f∗σ), (G, ς)).
Proof. The proof is the same as in the general case where Real structures are not concerned (see for instance [10, Proposition 2.3.3]). Definition 2.9. Given a continuous Real map f : (Y, %) −→ (X, ρ) and Real sheaves (F, σ) and (G, ς) as above, we define the set Homf(F, G)σ,ς of
Real f -morphisms from (F, σ) to (G, ς) to be
HomShρ(X)((F, σ), (f∗G, f∗ς)) = HomSh%(Y )((f∗F, f∗σ), (G, ς)) .
Definition 2.10. Let (X•, ρ•) be a Real simplicial (resp. pre-simplicial)
space. A Real sheaf on (X•, ρ•) is a family (Fn, σn)n∈Nsuch that (Fn, σn) is a
Real sheaf on (Xn, ρn) for all n, and such that for each morphism f : [m] −→
[n] in ∆ (resp. ∆0) we are given Real ˜f -morphisms ˜f∗ ∈ Homf˜(Fm, Fn)σm,σn
such that
(2.15) f ◦ g]∗ = ˜f∗◦ ˜g∗, whenever f and g are composable.
One can use the definition of the push-forward to give a concrete inter-pretation of this definition. Roughly speaking, a sequence (Fn, σn)n∈N is a
Real sheaf on a Real simplicial (resp. pre-simplicial, . . . ) space (X•, ρ•), if
for a given morphism f : [m] −→ [n] in ∆ (resp. ∆0, . . . ), then for any pair of open sets U ⊂ Xn and V ⊂ Xm such that ˜f (U ) ⊂ V there is a restriction
map ˜f∗ : Fm(V ) −→ Fn(U ) such that the diagram
(2.16) Fm(V ) σm V ˜ f∗ // Fn(U ) σn U Fm(ρ(V )) ˜ f∗ // Fn(ρ(U ))
commutes, and ˜f∗◦ ˜g∗ = ]f ◦ g∗ : Fk(W ) −→ Fn(U ) whenever ˜g(V ) ⊂ W ⊂ Xk. Morphisms of Real sheaves over (X•, ρ•) are defined in the obvious way;
